A centering clamping apparatus of the aforementioned type is known from the as yet unpublished U.S. patent application Ser. No. 14/538,990. This centering clamping apparatus consists of a housing and a centering element mounted to be displaceable along a longitudinal axis between an open position and a clamping position. A finger element, mounted on the centering element, is displaceable transversely to the longitudinal axis between a retracted position and an extended position. The finger element is operatively connected with an adjusting member displaceably mounted in the centering element. Within the framework of the disclosure, the finger element is always provided with at least one finger element and two finger elements are actually provided in the above described solution.